R, 30 bytes

Or **R>=4.1, 23 bytes by replacing the word function with a \.

function(v)sum(v%%2)%in%c(0,2)

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The straightforward approach.

Less straightforward, but used by many other answers, approach results in the same byte-count:

function(v)abs(sum(v%%2)-1)==1

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R, 30 29 bytes

Or **R>=4.1, 22 bytes by replacing the word function with a \.

Edit: -1 byte thanks to @Giuseppe.

function(v)sum(v%%2/2)%in%0:1

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Almost straightforward approach.

Less straightforward, but used by many other answers, approach results in the same byte-count:

function(v)(sum(v%%2)-1)^2==1

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